The one-sheeted hyperboloid, level set S = F −1(0) of F(x1, x2, x3) = x2 1 a2 + x2 2 b2 − x2 3 c2 − 1, or the two-sheeted hyperboloid with F(x1, x2, x3) = −x2 1 a2 − x2 2 b2 + x2 3 c2 − 1, are regular...

The one-sheeted hyperboloid, level set S = F −1(0) of F(x1, x2, x3) = x2 1 a2 + x2 2 b2 − x2 3 c2 − 1, or the two-sheeted hyperboloid with F(x1, x2, x3) = −x2 1 a2 − x2 2 b2 + x2 3 c2 − 1, are regular surfaces. A parametric representation is given, respectively, by x(u, v)=(a cos v cosh u, b sin v cosh u, c sinh u), and x(u, v)=(a cos v sinh u, b sin v sinh u, c cosh u), where (u, v) ∈ R × [0, 2π].